To perform this analysis we extend the UFA15 framework by adding interactions with gas in the source environment (as described in Section REF ), as well as making a variety of technical improvements to the analysis in [1]}. Based on the results of [2]}, we adopt a source evolution following the star-formation rate (SFR, [3]}) and take a single-mass injection of CRs into the source environment.The flux of neutrinos observed at Earth is sensitive to the source evolution at high redshifts, however, we do not consider other source evolutions here as the SFR evolution was found in [2]} to give one of the best-fits to UHECR data. Source evolutions which are stronger at high redshift generally result in a larger neutrino flux at Earth, but the quality of
the fit to UHECR observations degrades and requires an extremely
hard CR spectrum escaping the source [2]}, [6]}. Source evolutions with lower source densities at high redshift produce smaller neutrino fluxes at Earth and also give a worse fit to UHECR data, so are not explored in this study. A more complex injection composition was found not to be needed [2]}, so for simplicity we adopt single-mass injection. We approximate the gas to be pure hydrogen since other components make up less than \(10\%\)  by number. We also introduce an alternate treatment of the energy dependence of the escape time, based on the behavior of diffusion coefficients and reflecting the finite size of sources. Details of our treatment of systematic uncertainties are given in Section .
